Verify the identity 1tan2x1cot2xcsc2xsec2xSolution1tan2x1cot

Verify the identity. 1/tan^2x-1/cot^2x=csc^2x-sec^2x

Solution

(1/tan²x)-(1/cot²x)

write tanx as sinx/cosx , cotx as cosx/sinx

=(1/(sinx/cosx)²)-(1/(cosx/sinx)²)

=(1/(sin²x/cos²x))-(1/(cos²x/sin²x))

=(cos²x/sin²x)-(sin²x/cos²x)

=(cos²x(1/sin²x))-(sin²x(1/cos²x))

=(cos²x(1/sinx)²)-(sin²x(1/cosx)²)

write sinx as 1/sinx as cscx , 1/cosx as secx

=(cos²x(cscx)²)-(sin²x(secx)²)

=(cos²xcsc²x)-(sin²xsec²x)

given identity is not true

(1/tan²x)-(1/cot²x)=(cos²xcsc²x)-(sin²xsec²x) is true